The Space of Rational Maps on P
نویسندگان
چکیده
The set of morphisms φ : P → P of degree d is parametrized by an affine open subset Ratd of P . We consider the action of SL2 on Ratd induced by the conjugation action of SL2 on rational maps; that is, f ∈ SL2 acts on φ via φ = f ◦ φ ◦ f . The quotient space Md = Ratd/SL2 arises very naturally in the study of discrete dynamical systems on P. We prove that Md exists as an affine integral scheme over Z, that M2 is isomorphic to A2Z, and that the natural completion of M2 obtained using geometric invariant theory is isomorphic to P2Z. These results, which generalize results of Milnor over C, should be useful for studying the arithmetic properties of dynamical systems.
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